What is (\partial)/(\partial x)(2/(x^2+y^2+z^2)) ?

The answer to (\partial)/(\partial x)(2/(x^2+y^2+z^2)) is -(4x)/((x^2+y^2+z^2)^2)

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(\partial)/(\partial x)(2/(x^2+y^2+z^2))

\frac{\partial}{\partial x} (\frac{2}{x ^{2} + y ^{2} + z ^{2}})

- \frac{4 x}{( x ^{2} + y ^{2} + z ^{2} ) ^{2}}

Solution steps

\frac{\partial}{\partial x} (\frac{2}{x ^{2} + y ^{2} + z ^{2}})

Treat

y, z

as constants

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 2 \frac{\partial}{\partial x} (\frac{1}{x ^{2} + y ^{2} + z ^{2}})

Apply exponent rule:

\frac{1}{a} = a^{- 1}

= 2 \frac{\partial}{\partial x} ((x^{2} + y^{2} + z^{2})^{- 1})

Apply the chain rule:

- \frac{1}{( x ^{2} + y ^{2} + z ^{2} ) ^{2}} \frac{\partial}{\partial x} (x^{2} + y^{2} + z^{2})

= - \frac{1}{( x ^{2} + y ^{2} + z ^{2} ) ^{2}} \frac{\partial}{\partial x} (x^{2} + y^{2} + z^{2})

\frac{\partial}{\partial x} (x^{2} + y^{2} + z^{2}) = 2 x

= 2 (- \frac{1}{( x ^{2} + y ^{2} + z ^{2} ) ^{2}} \cdot 2 x)

Simplify

2 (- \frac{1}{( x ^{2} + y ^{2} + z ^{2} ) ^{2}} \cdot 2 x) : - \frac{4 x}{( x ^{2} + y ^{2} + z ^{2} ) ^{2}}

= - \frac{4 x}{( x ^{2} + y ^{2} + z ^{2} ) ^{2}}

\int 20 x (5 x^{2} - 3)^{3} d x

\int (sin^{2} (x))^{2} d x

\frac{d ^{2} y}{d x ^{2}} + 4 y = 0

(f (x) (2 x))^{'}

x \to - 8 lim (x + 8)

What is (\partial)/(\partial x)(2/(x^2+y^2+z^2)) ?
The answer to (\partial)/(\partial x)(2/(x^2+y^2+z^2)) is -(4x)/((x^2+y^2+z^2)^2)